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For maximizing the objective function $\sum_i{d_i y_i}+ A x - B \cdot \mathbb{I}_{x>0}$, how can I linearize $ A x - B \cdot \mathbb{I}_{x>0}$ term where $d_i, A$ and $B$ are positive constants and $x, y_i$ are continuous non-negative variables? Can you explain the logic of linearization? There are many other constraints involving $y_i, x$.

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  • $\begingroup$ Minimize or maximize? $\endgroup$
    – RobPratt
    Sep 8 '20 at 22:43
  • $\begingroup$ @RobPratt Maximize $\endgroup$
    – Al Guy
    Sep 8 '20 at 22:53
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If $M$ is a (small) upper bound on $x$, introduce a binary variable $z$ and big-M constraint $x \le M z$. The idea is that $x>0$ implies $z=1$. Now use $B z$ in the objective.

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  • $\begingroup$ But x=0 doesn’t imply z=0. Is it handled by requirement of maximization of the objective? $\endgroup$
    – Al Guy
    Sep 9 '20 at 3:07
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    $\begingroup$ Yes, the objective will naturally drive that. $\endgroup$
    – RobPratt
    Sep 9 '20 at 3:12

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